the function x 2 approximates to 4 within the standard .5 . For ( 1.9 ) 2 = 3.61 and ( 2.1 ) 2 = 4.41 , and thus the required interval 1.9 to 2.1 , containing 2 not as an end-point, has been found. This example brings out the fact that statements about a function f ( x ) in the neighbourhood of a number a are distinct from statements about the value of f ( x ) when x = a . The production of an interval, throughout which the statement is true, is required. Thus the mere fact that 2 2 = 4 does not by itself justify us in saying that in the neighbourhood of 2 the function x 2 is equal to 4 . This statement would be untrue, because no interval can be produced with the required property. Also, the fact that 2 2 = 4 does not by itself justify us in saying that in the neighbourhood of 2 the function x 2 approximates to 4 within the standard .5 ; although as a matter of fact, the statement has just been proved to be true.
If we understand the preceding ideas, we understand the foundations of modern mathematics. We shall recur to analogous ideas in the chapter on Series, and again in the chapter on the Differential Calculus.
Meanwhile, we are now prepared to define "continuous functions." A function is "continuous" at a value of its argument, when in the neighbourhood of its values approximate to (i.e. to its value at ) within every standard of approximation.
This means that, whatever standard be chosen, in the neighbourhood of approximates to within the standard . For example, is continuous at the value of its argument, , because however be chosen we can always find an interval, which (i) contains not as an end-point, and (ii) is such that the values of for arguments lying within it approximate to (i.e. ) within the standard . Thus, suppose we choose the standard ; now , and , and both these numbers differ from by less than . Hence, within the interval to the values of approximate to within the standard . Similarly an interval can be produced for any other standard which we like to try.